Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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The existence of $f$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$ implies the existence of a real $k$-lipschitz function $g$ such that $g|_Y=f$.
Here is the problem from a book for metric spaces I'm trying to solve:
Let $f:Y\rightarrow\mathbb{R}$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$. Prove that there is a $k$-lipschitz function $...
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when do we say a metric space is quasi-invariant under a function?
A measure of a space that is equivalent to itself under "translations" of this space. More precisely: Let $(X,B)$
be a measurable space (that is, a set $X$
with a distinguished $ σ$
-algebra ...
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Hyperbolic metric space and Cayley graph of a group [closed]
The following definition is given in the book "Group Theory from a Geometrical Viewpoint"
Proposition 2.1. The following are equivalent for a geodesic metric space X.
(1) Triangles are slim....
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Problem about fixed points in a complete metric space
Let $(X,d)$ be a non-empty complete metric space and let $ f:X \rightarrow X$ be a function such that for each positive integer $n$ we have
(i) if $ d(x,y)<n+1$ then $d(f(x),f(y))<n$
(ii) if $d(...
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Prove that the usual metric and other metric induce the same topology
I am working on A course on Borel sets, by S.M. Srivastava. There is this problem I am working on that states the following:
Show that both the metrics $d_1$ and $d_2$ on $\mathbb{R^n}$ defined in 2.1....
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?
Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question.
Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
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Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]
If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space?
...
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Proving a the distance between Cauchy sequences converges [duplicate]
Assume we have two Cauchy sequences { $x_n$ } and {$y_n$} in the metric space $(X,d)$. Is it true that the sequence {$a_n$}$=d(x_n,y_n)$ is convergent in $\mathbb{R}$? Here is my try: $$$$
Since those ...
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Real Analysis Question about Limit points and ε-neighborhoods
The question says "Prove that a point $x$ is a limit point of a set $A$ iff every ε-neighborhood of $x$ intersects $A$ at some point other than $x$."
I am having trouble proving the reverse ...
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What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?
For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let
$$
\lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }.
$$
And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
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If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
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There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero [duplicate]
Here we take
$$
\mathbb{N} := \{ 1, 2, 3, \ldots \}.
$$
Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
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The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by
$$
d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2.
$$
Let $\mathbb{R}$ and $...
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The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them
Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that
$$
d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0}
$$
Here we have the following definitions:
For any (...
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Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]
Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
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Is there a maximum number of disjoint balls of fixed radius I can fit into a compact metric space?
Let $(X,d)$ be a compact metric space and fix $r>0$. By sequential compactness, one may not find an infinite number of disjoint $r$-balls (sets $B_r(x):=\{y \in X: d(x,y)<r\}$) in $X$ as this ...
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Connected Metric Spaces: Strategies
I am not really sure if my ideas in this topic are correct. Can anyone help me?
Finding the connected components of a metric space $X$.
Suppose there are two connected components $C_1, C_2$ of $X$. ...
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Convex combination of equidistant curves
Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
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Can any open set in $\mathbb{R}^d$ be countably union of closed sets
I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
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Is a metric/distance not a measure?
A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might ...
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Finishing the proof of the triangle inequality of Hausdorff metric
currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck.
Setup:
Take $(X,d)$ as metric space.
Denote by $C(X)$ the set of closed ...
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How to understand the Sobolev space defined by completion.
In page 16 of this book,
the author state:
For $1\leq p <\infty$, consider the normed space of all smooth functions $\phi \in \mathbb{R}^n$ such that
$$
\|\phi\|_{1,p} = \|\phi\|_p + \|\nabla \phi\...
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Composition of asymmetric contraction mappings [closed]
Let $(M,d)$ and $(N,q)$ be metric spaces.
The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$.
Similarly, the operator $J:N\...
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Convex cocompact representation of finitely generated groups
Let $\mathbb{H}^n$ be the hyperboloid model for hyperbolic space and $\text{Isom}(\mathbb H^n) = PO(n,1)$.
Let $\rho: \Gamma \rightarrow PO(n,1)$ be a representation of finitely generated group $\...
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Is my understanding of the definition of a metric space correct?
The definition of metric space that I am using is as follows:
Let $X$ be a nonempty set. A function $d:X\times X\to \Bbb R$ is said to be a metric or a distance function on $X$ if $d$ satisfies the ...
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Intersection of interiors of sets in a partition of $\mathbb{R}^d$
Let $\mathcal{Q}=\{Q_1,...,Q_n\}$ and $\mathcal{P}=\{P_1,...,P_m\}$ be partitions of $\mathbb{R}^d$ with $n<m$. Assume all $Q_k\in\mathcal{Q}$ and $P_i\in\mathcal{P}$ are close and convex sets. ...
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Which metrics (on vector spaces) can be induced?
Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$.
I ...
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Axiom of Choice in characterizing openness in subspace
Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$.
$E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$.
The forwards direction usually ...
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Does this increasing sequence of subsets of a bounded connected metric space $(X,d)$ terminates at some point at $X$?
This might be a silly question; this is where I'm stuck as to whether a metric-bounded set in a connected metric space is uniformity-bounded in the sense of Bourbaki. Let $(X,d)$ be a bounded ...
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Is every bounded connected metric space totally bounded?
A metric space $(X,d)$ is said to be bounded if it is equal to a ball $B(x,r)$ of it. It is said to be totally bounded if for all $\epsilon>0$ there is a finite covering of $X$ by $\epsilon$-balls. ...
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